Optimal Multi-Robot Path Planning on Graphs: Complete Algorithms and Effective Heuristics (1507.03290v1)

Abstract: We study the problem of optimal multi-robot path planning on graphs MPP over four distinct minimization objectives: the makespan (last arrival time), the maximum (single-robot traveled) distance, the total arrival time, and the total distance. In a related paper, we show that these objectives are distinct and NP-hard to optimize. In this work, we focus on efficiently algorithmic solutions for solving these optimal MPP problems. Toward this goal, we first establish a one-to-one solution mapping between MPP and network-flow. Based on this equivalence and integer linear programming (ILP), we design novel and complete algorithms for optimizing over each of the four objectives. In particular, our exact algorithm for computing optimal makespan solutions is a first such that is capable of solving extremely challenging problems with robot-vertex ratio as high as 100%. Then, we further improve the computational performance of these exact algorithms through the introduction of principled heuristics, at the expense of some optimality loss. The combination of ILP model based algorithms and the heuristics proves to be highly effective, allowing the computation of 1.x-optimal solutions for problems containing hundreds of robots, densely populated in the environment, often in just seconds.

• The paper introduces a novel mapping between multi-robot path planning and network flow problems using ILP models to optimize various performance metrics.
• The exact makespan optimization algorithm is demonstrated to solve dense instances with a 100% robot-vertex ratio effectively.
• Heuristics, including a k-way split approach, significantly enhance computational efficiency by delivering near-optimal solutions for large-scale puzzles.

Overview of Optimal Multi-Robot Path Planning on Graphs: Complete Algorithms and Effective Heuristics

The paper "Optimal Multi-Robot Path Planning on Graphs: Complete Algorithms and Effective Heuristics" by Jingjin Yu and Steven M. LaValle tackles the complex problem of multi-robot path planning on graphs (MPP), a topic of significant relevance in the field of robotics and AI. The problem involves planning the paths of multiple distinguishable robots on a graph, aiming at minimizing metrics such as makespan, maximum single-robot traveled distance, total arrival time, and total distance. Recognizing the distinct nature of these objectives and their NP-hardness, the authors propose efficient algorithmic solutions that combine complete algorithms with principled heuristics.

Key Contributions

The paper introduces a pioneering one-to-one mapping between MPP and network-flow problems through integer linear programming (ILP) models, facilitating the derivation of algorithms capable of handling each of the four optimization objectives. A standout feature is the exact algorithm for makespan optimization, which for the first time, is demonstrated to solve highly dense instances with a robot-vertex ratio up to 100%.

For cases where slight optimality loss is acceptable, the research further refines these algorithms with multiple heuristics that dramatically enhance computational efficiency. Notably, the paper presents a $k$-way split heuristic, which divides the MPP problem over time, enabling the computation of $1.x$-optimal solutions for large, densely populated instances effectively and often within seconds.

Numerical Results and Implications

The numerical results highlight the impressive computational performance of the proposed methods. Particularly in scenarios such as the classic $N^2$-puzzle, the presented algorithms outperform conventional heuristic approaches like those in warehouse automation settings by a significant margin, handling up to 25-puzzle instances efficiently.

The implications of these results are far-reaching. Practically, this work facilitates the development of more efficient autonomous systems in areas such as warehouse logistics, where tasks can be completed faster and with reduced energy consumption. Theoretically, it opens avenues for further exploration of ILP-based techniques and their application to other domains requiring combinatorial optimization under constraints.

Speculation on Future Developments

Looking forward, these methodological advancements could pave the way for new exploration in cooperative robotics, potentially improving throughput in real-time applications like automated vehicles and drone fleet management. Additionally, given the generality of the ILP framework, there is potential to extend these algorithms to continuous domains or more complex environments with dynamic obstacles, further broadening their applicability.

In conclusion, this paper offers substantial contributions to the field of multi-robot path planning by rigorously addressing optimality challenges and proposing scalable, efficient solutions. The intersection of novel theoretical insights and practical efficiency marks a significant advancement in the understanding and application of multi-robot systems.